Coherent and semiclassical states of a free particle

Abstract

Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states.

Coherent and Semiclassical States of a Charged Particle in Electromagnetic Fields

“…The function (76) is a product of three one-dimensional GCS that depend on q 1 , q 2 and q 3 respectively. These one-dimensional GCS coincide with ones obtained and studied in our previous publications [12].…”

“…(6), while diverges according to Eqs. (12), due to the ω −1 singularity. At a first sight, this might be source of discrepancy but, to solve the Cauchy problem, one has to specify the initial coordinates x 0⊥ and velocities v 0 ⊥ .…”

“…In the frame of such a definition one can in principle construct CS for a general quadratic system. In the article [10] we, following these scheme, constructed different families of CS (GCS) for one-dimensional systems with general time-dependent quadratic Hamiltonian, see too [11,12]. In the article [12], we have demonstrated that the GCS of a free particle can be treated as quantum states that describe a semiclassical motion, whereas there exist GCS that describe pure quantum motion.…”

Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

“…which has been already reported in [77], p. 83. The above procedure permits the construction of coherent states for the free-particle system by means of a simple mapping of the Glauber states to the appropriate basis (similar results can be found in [79]). In such case, the function σ is proportional to the width of the wave-packet which, from (61), is an increasing function in time.…”

Coherent and Semiclassical States of a Charged Particle in a Constant Electric Field

“…It should be noted that the given criterion is not general. One of the adopted definitions follows from the inequality ζ ≫ 1, which is useful both for a particle moving in more than one dimension and for a particle in the so-called magnetic solenoid field[17])…”